My tech guy gave me a very simple workaround that solved this problem for me. I tried many other things people suggested until this worked. What a hero.
In Microsoft AutoUpdate, select an item and click the 'Update' button (not 'Update All'). While this item is downloading and you see the progress bar, hit the pause button before it downloads entirely, wait just a second, then unpause. My tech person did this once or twice during each individual download, then let the item finish downloading. It should then update without the error—at least it did, miraculously, for me.
After doing this with all the current updates in the queue, he had me close Microsoft AutoUpdate before refreshing it to check again for each new set of missed updates. He then had me click on those missed ones individually (I think I stopped hitting pause at that point). Each time we ran out of new updates, we closed Autoupdate again, then reopened it, before refreshing the next set of updates. Not sure whether that was necessary, but it worked.
This error drove me nuts, so I thought I'd record it for posterity here.